Chapter 2.1 - Groups, Fields and Vector Spaces

Wicker introduces a real grab-bag of mathematical ideas in this section.  Here are my notes.

Sets

In a set, only one instance of each distinct element is considered to be part of the set (duplicates are ignored), so the sets S = {1,2,3} and S = {1,2,1,1,3,3,2,1,2} are considered to be the same.

In a set, the order of the elements is not important, so the sets S = {1,2,3} and S = {1,3,2} are considered to be the same.

A set in which multiplicy and order are important is called a list.

Binary Operation

A binary operation is described as binary because it operates on two elements of a set to produce a third element in the same set (and not - as a digital designer might erroneously conclude - because it operates on zeros and ones).  For example 3 + 4 = 7 is an example of the binary operator "+" applied to the set of integers.

It is part of the definition of a binary operation on two elements of a set that the result is another member of the same set.  This property is called "closure".

In the general case a binary operation is not necessarily commutative.  In other words, if we represent the binary operation by the symbol "·" then the set element produced by a · b is not necessarily the same one as produced by b · a.  Consider as an example of this, the binary operation of matrix multiplication applied to the set of 2 x 2 matrices,

So, usually, a binary operation is defined as operating on an ordered pair of set elements to explicitly exclude any requirement for commutativity.

The "not necessarily distinct" comment about the result of a binary operation simply means that the result might be the same as one of the input elements, e.g. 3 + 0 = 3, or that two different pairs might produce the same result 3 + 4 = 7 and 2 + 5 = 7.

Group

A set operated on by a binary operation is a very primitive algebraic structure.  There's only one rule, or axiom, that must be met (closure), and that comes "for free" as part of the definition of a binary operation.  Nonetheless, this structure has a name, it's called a groupoid.

If we also require that the binary operation is associative then we have defined an associative groupoid, which is also known as a semigroup. If we further require that the semigroup contains an identity element (and therefore cannot be empty) then we have defined a monoid.  Finally, if we also require that every element of the monoid must have an inverse, then we have defined a group.

The point of calling out all these intermediate structures is to demonstrate that a group is just a set over which we have defined a binary operation that also conforms under all circumstances to a selection of additional rules or axioms.

This is, however, a rather deprecating description of a group.  The (carefully) chosen rules are found to be fundamental to an enormous variety of mathematical and physical systems (especially those involving symmetry). So the "set and binary operator and axioms" algebraic structure that we call a group, while in some sense an arbitrary mathematical "invention", proves to have widespread applicability.  Any mathematical system that is found to have group properties can immediately take advantage of the enormous body of group theory that has been built on top of these basic axioms.

So if, when applied to the elements of a set G, the binary operation "·" satisfies the following four rules (the group axioms), then the combination of set G and binary operation "·" is a group.

1. Closure: For all a, b Î G, a · b Î G
2. Associativity: For all a, b, c Î G, (a · b) · c = a · (b · c)
3. Identity: There exists an element e Î G such that for all a Î G,  e · a = a = a · e
4. Inverse: For each a Î G there exists a ÎG such that a · (-a) = e = (-a) · a

If the additional commutativity axiom is satisfied then the group is a commutative group or (equivalently) an abelian group.

5. Commutativity: For all a, b Î G, a · b = b · a

It's an easy convenience, and not uncommon, to think of the group operation as multiplication, but it can, in fact, be any operation that complies with the group axioms.  It could, for example, represent addition under modular arithmetic rules or a composition of symmetries.

If it is helpful, we can convey the nature of the group operation by talking about, for example, multiplicative groups, additive groups or symmetry groups.  However, it's worth remembering that regardless of the mechanics of the group operation, it's the isomorphism class of the end result that really defines which group we are dealing with.

I personally don't agree with Wicker's suggestion that commutative groups are in some sense synonymous with additive groups.  There are many important examples of multiplicative commutative groups. See, for example, Wicker's theorem 2-2.

Modular Arithmetic

It's not a big deal, but Wicker plays a little fast and loose with the distinction between congruence and equivalence.  To his credit he does define what he means by "equivalence" and the definition he gives is essentially that of congruence.

When we divided 13 by 5 we produced two results; the integer part, 2, (sometimes called the quotient) which is the number of times the divisor, 5, "goes into" 13; and the remainder 3.

In modular arithmetic we are not so interested in the integer part, but we are very interested in the set of possible remainders (residues) for a given divisor (modulus) and in particular how the properties of this set are affected by the choice of modulus.  The terms residue and modulus are used, instead of the more generic "remainder" and "divisor", to identify and convey precise meaning in the context of modular arithmetic.

Let's think about dividing by 5 for a little longer.  If we divide any of the numbers in the infinite sequence ... -17, -12, -7, -2, 3, 8, 13, 18, 23,... by 5, the residue is always 3.

Taking 18 as an example, we would write,

18 ≡ 3 mod 5

Evidently, there is some kind of relationship between these numbers when they are divided by 5 because they all have residue 3.  Formally, we say that they are congruent modulo 5.  If we accept Wicker's blurring of congruence and equivalence then we can see that elements of the set {...-17, -12, -7, -2, 3, 8, 13, 18, 23,...} are congruent or "equivalent" under modulo 5 addition because (as Wicker says) you could interchangeably use any one of these numbers in a modulo 5 sum and you'd get the same answer.

One way of thinking about this "equivalence" is to imagine a tape measure wrapped round a pole of circumference 5.  Then every fifth number on the number line (tape measure) marks the same point on the circumference of the pole...

...and so we have five infinite sets of congruent or "equivalent" integers (I've included the negative integers, to be a little more theoretically rigorous).

{...,-20, -15, -10, -5, 0, 5, 10, 15, 20,...}
{...,-19, -14, -9, -4, 1, 6, 11, 16, 21,...}
{...,-18, -13, -8, -3, 2, 7, 12, 17, 22,...}
{...,-17, -12, -7, -2, 3, 8, 13, 18, 23,...}
{...,-16, -11, -6, -1, 4, 9, 14, 19, 24,...}

These infinite sets are called equivalence classes.   The smallest non-negative number less than the modulus is called the common residue (that is, the residue common to all members of the equivalence class).

It is conventional (and convenient) to use the common residue as the class representative, which we can write as, for example, [3] to remind us that it is representing a set, or simply 3 if we don't care.

The set of all the possible residues for a given modulus, in other words the set of equivalence class representatives, which in this case is {0,1,2,3,4}, is called the residue class.

Order of a Group Element

The notation used here, for example g³ = g · g · g, is suggestive of raising g to a power, but it's important to realise that its just a convenient shorthand for repeated application of the group operation to the element g.  For example, if the group operation was addition then g³ = g + g + g.  Which is somewhat contrary to what that notation would lead one to expect in "normal" arithmetic!

If the group operation is multiplicative, then the identity is 1 and the element order is given by how many times the element is used in order to "get back to" 1.

If the group operation is additive, then the identity is 0 and the element order is given by how many times the element is used in order to "get back to" 0.

So to elaborate on Wicker's example 2-7 where he investigates the orders of the elements of the group {1,2,3,4,5,6} under modulo 7 multiplication...

1 = identity, one use so order is 1
2 x 2 = 4, 4 x 2 = 8 ≡ 1, three uses so order is 3
3 x 3 = 9 ≡ 2, 2 x 3 = 6, 6 x 3 = 18 ≡ 4, 4 x 3 = 12 ≡ 5, 5 x 3 = 15 ≡ 1, six uses so order = 6
4 x 4 = 16 ≡ 2, 2 x 4 = 8 ≡ 1, three uses so order = 3
5 x 5 = 25 ≡ 4, 4 x 5 = 20 ≡ 6, 6 x 5 = 30 ≡ 2, 2 x 5 = 10 ≡ 3, 3 x 5 = 15 ≡ 1, six uses so order = 6
6 x 6 = 36 ≡ 1, two uses so order = 2

Subgroups

Wicker uses the elegant minimalism of "if for all a and b in H, c = a · b^-1 is also in H, then H is a subgroup" to define a subgroup.  This neatly encapsulates a requirement for closure, inverses and identity, but I have to say it's not the most transparent way of introducing the concept of a subgroup.

The proof that this test does indeed define a subgroup is given, for example, as the proof of Theorem 3.1 on page 58 of Contemporary Abstract Algebra by Joseph A. Gallian.

Here's another more plodding approach to the same idea.

Imagine we have a set which we know to be a group under the binary operation "·".  In fact, let's take as a convenient example the group G = {1,2,3,4,5,6} under modulo 7 multiplication.

If there is a subset of this group that also satisfies the group axioms under the same operation (in this case, modulo 7 multiplication) then it is a subgroup.

There are many ways you can find subgroups ranging from pure guesswork to judicious application of the Fundamental Theorem of Galois Theory, at this stage let's do some (educated) guesswork...

Consider the subset H = {1,2,4}... does it have what it takes?

1. Closure: For such a small set we can evaluate this exhaustively to confirm that it is closed under modulo 7 multiplication.

2. Associativity: We don't need to check this property for the subset H because we know G is associative.  This property is "inherited".
3. Identity: G contains the unique identity element 1 so this must be the indentity element for H too (because if H Ì G had a different identity element, then G would have two identity elements) so we only need to check that 1 Î H, which it is.
4. Inverse: We need to check that for each a Î H we have a ^-1 Î H.  We know a ^-1 exists because it's in G, we just have to check it's also in H.  Fortunately, 2 = 4 ^-1 and 4 = 2 ^-1 under modulo 7 multiplication.

We conclude that H = {1,2,4} is a subgroup of G = {1,2,3,4,5,6} under modulo 7 multiplication.

Returning for a moment, to Wicker's definition; "if for all a and b in H, c = a · b^-1 is also in H, then H is a subgroup".  This is simply a neat embodiment of the three relevant tests (identity, inverses and closure) in one expression.  See Gallian for the details.

K = {1,6} is also a subgroup of G.

Finally, Wicker mentions proper subgroups.  The subsets of G include G itself (because subset implies "Í" rather than the more strict condition "Ì"), so we classify G itself as one of the subgroups of G, but its not a proper subgroup because it is not smaller than G.

Left and Right Cosets

We'll introduce the basic idea of cosets, as Wicker does, with an example of the cosets of a subgroup of a finite group.

Let's take as an example the group G = {1,2,3,4,5,6} under modulo 7 multiplication.  We know from before that K = {1,6} is a subgroup of G, but what about the members of G that that leaves behind, namely 2, 3, 4 and 5, is there some complementary way we can arrange them too?

Well we could take the subgroup H = {1,2,4} instead, that would leave 3, 5 and 6 behind, but we still haven't figured out how to arrange those leftovers.  One thing is clear, once we've "used up" the identity element,1, to make a subgroup, we cannot make another distinct subgroup with what's left.

That's where cosets come in.  Take the subgroup K and for each element in K, multiply it by 2, 3, 4 and 5

2 x 1 = 2 mod 7, 2 x 6 = 5 mod 7

3 x 1 = 3 mod 7, 3 x 6 = 4 mod 7

4 x 1 = 4 mod 7, 4 x 6 = 3 mod 7

5 x 1 = 5 mod 7, 5 x 6 = 2 mod 7

What we find is that the elements of G that are not in the subgroup K fall into subsets {2,5} and {3,4} that are the same size as K with the property that

{2,5} x {1,6} = {2,5}

{3,4} x {1,6} = {3,4}

These subsets are called cosets and have the interesting property that they also form a group, in this case the finite group of order 3 isomorphic to C₃.

Next show that 5Z is an (infinite) subgroup of Z and that 5Z+1, 5Z+2, 5Z+3 and 5Z+4 are the (infinite) cosets of 5Z.  They are also the equivalence classes modulo 5 and they form a group, which you can think of as a group of cosets, or as the group Z5 with the elements being the (simpler) class representatives.

Note {1,6} is considered to be a coset as well as a subgroup.

Note commutative group operations mean left and right cosets are the same.

Not mentioning here that Z is a ring, or that 5Z is an ideal, or Z/5Z is a quotient ring isomorphic to Z5.

Cosets are disjoint, cosets are all the same size => |subgroup| must divide |group| (Lagrange).

Wicker doesn't do much more at this stage than say what cosets are.  It appears to me that he mainly introduces them to serve as a stepping stone to Lagrange's theorem.

Lagrange's Theorem

Rings

A ring is the next major step up the evolutionary ladder of algebraic structures.  Again we start with a set R (which can be finite or infinite), but this time we define two binary operators.  We also get a bit less abstract and actually represent them by the symbols, + and × , which we commonly interpret as addition and multiplication.  We also extend the set of rules (axioms) to include...

1. Additive closure: For all a, b Î R, a + b Î R
2. Additive associativity: For all a, b, c Î R, (a + b) + c = a + (b + c)
3. Additive identity: There exists an element 0 Î R such that for all a Î R,  0 + a = a = a + 0
4. Additive inverse: For each a Î R there exists -a Î R such that a + (-a) = 0 = (-a) + a
5. Additive commutativity: For all a, b Î R, a + b = b + a

So the structure is an abelian group under addition.  Note that we've represented the additive identity as 0 as well, rather than the more generic "e".

We then include,

6. Multiplicative closure: For all a, b Î R, a × b Î R
7. Multiplicative associativity: For all a, b, c Î R, (a × b) × c = a × (b × c)

So the structure is also a semigroup under multiplication.  Finally, we bind these two sets of rules together with an eighth rule,

8. Left and right distributivity: For all a, b, c Î R, a × (b + c) = (a × b) + (a × c) and (b + c) × a = (b × a) + (c × a)

This constitutes a ring.

In rule 8 we have specified left and right distributivity because we have not specified multiplicative commutativity.  Nor, you may notice, have we specfied multiplicative identity.

A ring is therefore an abelian group under addition and a semigroup under multiplication.

If we also specify multiplicative commutativity then we define a commutative ring.

If we also specify multiplicative identity then we have a commutative ring with identity.

If we also exclude zero divisors then it becomes an integral domain.

If we add multiplicative inverse then we get a field.

Fields

Vector Spaces

Spanning Sets

Bases

Dimension

The Inner Product

Dual Spaces of Vector Spaces

The definition of a dual space given by Wicker is undoubtedly rigorous, but I didn't feel it gave me much insight into what a dual space actually is.  Here's my rather imprecise description to perhaps shed some light on the rigorous definitions.

If I have a vector space V of dimension n from which I have selected a set of k linearly independent vectors to serve as the basis for a vector subspace S of dimension k, then the dual space S^{^} is the set of vectors in V that are orthogonal to all vectors in S.

That is essentially what Wicker's definition 2-11 says.  But what that also implies is that no matter which set of k vectors you choose as the basis for S, the members of S^{^} do not project onto any of those basis vectors (because we exhaustively checked the dot products of each member of V against each member of S to identify S^{^} in the first place).  So the members of S^{^} must have a basis in S^{^} and those basis vectors (which are also members of V) must be linearly independent of the basis vectors of S.  Since we know the dimension of V is n and the dimension of S is k, the dimension of S^{^} can at most be n-k because if we add n-k vectors from S^{^}, each (by definition) linearly independent to the k vectors in the basis of S, then we have created an n-dimensional basis which must be a spanning set for V.  Wicker proves that the dimension of S^{^} is exactly n-k.

So... if I wanted to add n-k basis vectors to the set of k vectors that serve as a basis for S so that I again had a basis for V, then I would have to draw them from the dual space ; in other words, the dual space is also the “pool” of candidate basis extending vectors.

The Dimension Theorem